318 digital compensation zoh vogel krall 318

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Ubiquitous Computing and Communication Journal 1

DIGITAL COMPENSATION OF IN-BAND IMAGE SIGNALS CAUSED

BY M-PERIODIC NONUNIFORM ZERO-ORDER HOLD SIGNALS

Christian Vogel1, Christoph Krall

2

1

ETH Zurich, SwitzerlandSignal and Information Processing Laboratory (ISI)

[email protected] & Schwarz GmbH & Co. KG

Radiocommunications Systems Division

[email protected]

ABSTRACT

In this paper we introduce the design of a digital time-varying filter to compensate

the spurious in-band spectra caused by M-periodic nonuniform zero-order hold

signals. For this purpose, a continuous-time framework to describe and analyze

such signals is developed. Based on the error analysis, an equivalent discrete-timeframework is derived and used for the design of a discrete-time time-varying

precompensation filter. We exemplify the design of the precompensation filter for

the two-periodic case through simulations and measurements. Furthermore, we

discuss Farrow filters to reduce the design complexity.

Keywords: periodic, nonuniform, zero-order hold, compensation, reconstruction.

1 INTRODUCTION

Emerging communication systems require

high-speed and high-resolution data converters [1].

Since the integration density of digital circuitsgrows much faster than the one of analog circuits,

we have an increasing gap between analog and

digital circuits in terms of speed, area, and power

consumption. In particular, data converters - as the

central interface between analog and digital circuits

- are affected by this technological gap [2]. A new

design paradigm exploits the technology gap by

using digital signal processing to overcome the

analog impairments of data converters, e.g. [3].

In this paper we discuss digital-to-analog

converters (DACs) using a zero-order hold (ZOH)

for the signal reconstruction. We investigate the

compensation of in-band images caused by ZOHswith periodic nonuniform hold signals, i.e., a

nonuniform ZOH, as illustrated in Fig. 1. The

individual sample instants deviate by rnT from the

ideal time instants nT , where T is the nominal

sampling period of the DAC and r n

is the relative

time offset for time n with period M , i.e.,

rn= r

n+ M . Beside the typical sin x ( ) / x shaped

output spectrum, such nonuniform ZOH signals

introduce additional spurious images that

significantly reduce the DAC performance [4]. We

can find such spurious images in DACs driven by

clock signals with deterministic jitter [5] and intime-interleaved DACs with timing mismatches [6-

8]. Although the effects of

y(t)

x(0T )

x(1T )x(2T )

x(4T )

0T + r0T 1T + r1T

2T + r2T 3T + r0T

4T + r1T t

x(3T )

Figure 1: Output of a three-periodic nonuniform

ZOH (M=3) in the time domain.

nonuniform ZOHs have been extensively analyzed

[4,5,9], their digital compensation has not yet been

addressed in the literature.

We propose a digital time-varying filter to

compensate for the spurious image spectra caused

by periodic nonuniform ZOH signals. Therefore wedevelop a discrete-time model of a nonuniform

ZOH in Sec. 3 and use this model in Sec. 4 to

derive the ideal frequency response of a time-

varying compensation filter. In Sec. 5 we present

filter design examples including a Farrow filter

design and finally draw our conclusions in Sec. 6.

We want to note that preliminary results of the

presented work have been presented at two

conferences [10,11].

2 PRELIMINARIES

Sampling a continuous-time signal x c (t ) atuniform time instances t = nT produces the




sequence x[n]= xc(nT ), where T denotes the

sampling period. The discrete-time Fourier

transform of x [n] is

X e j ( ) =

1

T X c j

T p

2

T

p=

. (1)

Throughout this paper, we assume that the signal

x c(t ) is bandlimited, i.e., the continuous-time

Fourier transform of x c(t ) satisfies

X c j ( ) = 0, for

T . (2)

With these assumptions, the signal x c(t ) can be

recovered from the sequence x [n] by using a

sequence-to-impulse train converter and an ideal

continuous-time low-pass filter with gain T and

cut-off frequency /T .

y(t)

x(4T )

x(3T )

x(1T )

x(0T )

0T 1T 2T 3T 4T t

x(2T )

Figure 2: Ideal uniform ZOH.

In practice, however, we can neither generateideal Dirac-delta impulses nor can we implement an

ideal continuous-time low-pass filter. Instead a

ZOH and a low-order analog low-pass filter are

used. Employing an ideal uniform ZOH for the

reconstruction process results in the reconstructed

signal y(t ) , which is illustrated in Fig. 2 and is

given by

y(t ) = x0 (t ) h0 (t )T (3)

where

x0 (t ) = xc(t ) t nT ( )

n=

(4)

is the sampled continuous-time signal and

h0(t ) =

1

T u t ( ) u t T ( )( ) (5)

is the 1/T -weighted impulse response of an ideal

ZOH with u(t ) being the unit step function. The

reason to explicitly introduce the T -factor in

Eq. (3) can be seen in the frequency domain.

Applying the continuous-time Fourier transform(CTFT) to Eq. (3) results in

Y ( j ) =1

T X c j p

2

T

H 0 j ( )

p=

T

= X c j p2

T

H 0 j ( )

p=

(6)

with

H 0 ( j ) =

sin T

2

T

2

e j

T

2 . (7)

We realize that the factor T of the ZOH

compensates the 1/T -factor introduced by the

sampling process. Furthermore, the output signal

consists of the 2 /T -periodic extended

continuous-time signal X c ( j ) weighted by the

sinc( x ) function given by Eq. (7).

3 SYSTEM MODELS

After clarifying the behavior of a uniform ZOH,

we introduce a ZOH with M -periodic nonuniform

hold signals and derive a discrete-time model.

3.1 Continuous-time model

Figure 1 illustrates the output of a ZOH with

M -periodic nonuniform hold signals in the time

domain. In contrast to the uniform case, each

sampling instant is time-shifted by r n, where r

nis

periodic with M , i.e., rn= r

n+ M . These time offsets

of the hold signal can be modeled by the M -

periodic impulse response hn= h

n+ M starting at r

n

and ending at (1+ rn+1

)T [4,5], i.e.,

hn(t ) =

1

T u t r

n( ) u t (1+ rn+1

)T ( )( ) . (8)

Therefore, the output signal y t ( ) in Fig. 1 can be

modeled as the sum of M weighted impulse trains

of period MT

xn(t ) = x

c(t ) t mMT nT ( )

m=

(9)

convolved with M impulse responses hn(t )T , i.e.,

y(t ) = xn (t ) hn (t )T n=0

M 1

. (10)

The CTFT of Eq. (10) results in




Y ( j ) = X c j k 2

MT p

2

T

k =0

M 1

p=

(

H k j ( )

(11)

with

(

H k j ( ) =1

M H n j ( )

n=0

M 1

e jkn

2

M (12)

and

H n ( j ) =

sin 1+ rn+1 rn( )T

2

T

2

e j 1+rn+1+rn( )

T

2 . (13)

0 0.5 1 1.5 2 2.5 3−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

E n e r g y D e n s i t y S p e c t r u m ( d

B c )

Normalized frequency (Ω/ Ωs)

Desired

Image 1Image 2

Image 3

Figure 3: Output of a DAC with a 4-periodic

nonuniform ZOH (M=4) in the frequency

domain.

Figure 3 illustrates the spectral components

given by Eq. (11)-(13) for M = 4 . The desired

output is the 2 -extended continuous-frequency

spectrum weighted by

(

H 0( j ), which is the average

frequency response over all frequency responses

H n ( j ) for n = 0,..., M 1. This spectrum is

similar to the output spectrum in the uniform case;

however, we additionally have M 1 spectra,

which are modulated and filtered images of the

spectrum of a uniform ZOH. As these images

appear within the fundamental band, i.e., < /T ,

they cannot be removed by an analog low-pass

filter and significantly degrade the output signal

quality. A detailed analysis of the impact on the

performance can be found in [4,5].

3.2 Discrete-time model

In order to derive discrete-time compensation

filters for the in-band images, we represent the

continuous-time output signal given by Eq. (11) for

< /T in discrete-time. Therefore, the output in

Eq. (11) has to be ideally bandlimited, which can be

realized by the filter

H id j ( ) =1, <

T

0,

T

. (14)

Using Eq. (14) and Eq. (11), we can express the

ideally bandlimited signal as

Y b( j ) =1

T Y ( j )TH id ( j ) . (15)

Because we assume that the underlying continuous-

time signal x c(t ) is bandlimited, which is

implicitly satisfied if x [n] is a synthetic signal, we

can relate [12, Chap. 4]

X e j ( ) =

1

T X c j

T

(16)

and

(

H k e j ( ) =

(

H k j

T

(17)

for < . Consequently, we can express Eq. (11)-

(13) in discrete-time as

Y (e j

) = X e j

k

2

M

(

H k e j

( )k =

(18)

with

(

H k e j ( ) =

1

M H n e

j ( )n=0

M 1

e jkn

2

M , (19)

and

H n (e j

) =

sin 1+ rn+1 rn( )1

2

1

2

e

j 1+rn+1+rn( )1

2 . (20)

Using Eq. (18) and Eq. (1) with Y (e j

) it can be

verified that

Y b( j ) =Y (e j T

)TH id ( j ) (21)

results in Eq. (15) and the continuous-time and the

discrete-time frequency output are related by

Y (e

j

)=

1

T Y j

T

, <

. (22)




4 DERIVATION OF THE COMPENSATION

FILTER

In this section we propose a time-varying filter

to compensate the in-band images. We derive the

design equations and analytically solve them for the

two-periodic case.

4.1 Design equationsTo compensate for the spurious images, we

propose an M -periodic time-varying filter gn[ l ] ,

which relates the input x [n] to the output s[n] as

s[n]= gn[ l ] x[n l ]l =

. (23)

The precompensation filter and the discrete-time

model of the nonuniform ZOH are illustrated in

Fig. 4 for the two-periodic case.

discrete-time model ZOH

H 1(e jω)

(−1)n

Gn(e jω)

(−1)n

H 0(e jω)

G1(e jω)

G0(e jω)s[n]x[n] y[n]

Figure 4: Precompensation filter in front of the

discrete-time model of a nonuniform ZOH

(M=2).

The time-varying filter gn[ l ] should modify the

signal x [n] in such a way that the signal s[n]

driving the nonuniform ZOH does not generate

image signals within the fundamental band.

Since gn[ l ] is periodic with M , we can

represent it as the inverse discrete-time Fourier

series (DTFS) [13, Chap. 4]

gn[ l ]=(

gk [l ]k =0

M 1

e jkn

2

M (24)

with the related DTFS

(

gk [l ]=1

M gn[ l ]

n=0

M 1

e jkn

2

M . (25)

After substituting Eq. (24) in Eq. (23), we can write

s[n] =(

gk

k =0

M 1

[l ]e jkn

2

M x[n l ]l =

=

(

gk [l ] x[n l ]l =

e

jkn2

M

k =0

M 1

.

(26)

The discrete-time Fourier transform (DTFT) of

Eq. (26) gives

S e j ( ) =

(

Gk

k =0

M 1

e j k

2

M

X e

j k 2

M

(27)

where

(

Gk e j ( ) =

1

M Gn

n=0

M 1

e j ( )e

jkn2

M (28)

is the DTFT of Eq. (25). Because of the time-

varying filter gn[ l ], the input signal into the ZOH

is s[n] instead of x [n] and Eq. (18) becomes

Y (e j

) = S e j k

2

M

(

H k e j ( )

k =0

M 1

. (29)

Substituting the output of the filter given by

Eq. (27) in Eq. (29) results in

Y e j ( ) =

(

H k 1

e j ( )

(

Gk e j k

1+k ( )

2

M

k =0

M 1

k 1=0

M 1

X e j k

1+k ( )

2

M

.

(30)

With l = k 1+ k we can simplify Eq. (30) to

Y e j ( ) =

(

H l k e j ( )

(

Gk e j l

2

M

l =k

M 1+k

k =0

M 1

X e j l

2

M

=

(

H l k e j ( )

(

Gk e j l

2

M

l =0

M 1

k =0

M 1

X e j l

2

M

(31)

where we have exploited the periodicity of (

H l k

(e j

) .

Thus, the overall transfer function can be

expressed as

Y e j ( ) =

(

F l e j l

2

M

X e

j l 2

M

l =0

M 1

(32)

with




(

F l e j ( ) =

(

H l k k =0

M 1

e j +l

2

M

(

Gk e j ( ) (33)

as it is shown in Fig. 5 for the two-periodic case.

F 0(e jω)

F 1(e jω)

(−1)n

y[n]

x[n]

Figure 5: Overall transfer function of the

cascaded system for M=2.

From the theory of multi-rate systems we know

that a system is defined as perfect reconstruction

system if the output is a scaled and delayed version

of the input [14]. In our compensation problem wewant to maintain unity gain of the overall system

and therefore require that

(

F l e j l

2

M

=

e j

, for l = 0

0, for l =1,2,K, M 1

(34)

where is the delay of the system. If Eq. (34) is

fulfilled all image spectra are cancelled and the

sin( x ) / x distortions in the fundamental band are

equalized. The time-varying filter gn[ l ] can be

efficiently implemented as an M -channel

maximally decimated multi-rate filter bank [14].

4.2 Two-periodic nonuniform ZOHWe will exemplify the filter design procedure

for the two-periodic case, i.e., M = 2. By

expressing Eq. (33), and Eq. (34) in matrix notation

we obtain

e j

0

=

(

H 0e j ( )

(

H 1e j ( )

(

H 1e j + ( )( )

(

H 0e j + ( )( )

(

G0e j ( )

(

G1e j ( )

(35)

Solving the matrix equation leads to

(

G0

e j ( )

(

G1

e j ( )

=

(

H 0

e j + ( )( )

(

P e j ( )

(

H 1

e j + ( )( )

(

P e j ( )

e j

(36)

with

(

P e j ( ) =

(

H 1

e j ( )

(

H 1

e j + ( )( )

(

H 0

e j ( )

(

H 0

e j + ( )( ).

(37)

Applying the inverse DTFS as defined in Eq. (24)

to Eq. (36) and substituting the relation Eq. (19) for (

H k (e j

) we obtain the transfer functions of the

time-varying filter gn[ l ] that are

G0

e j ( )

G1

e j ( )

=

H 1 e

j + ( )

( )P e j ( )

H 0

e j + ( )( )

P e j ( )

e j

(38)

with

P e j ( ) =

1

2 H

1e

j ( ) H 0

e j + ( )( )( H

0e

j ( ) H 1

e j + ( )( )).

(39)

Analytically solving the matrix equations for larger orders of M is infeasible. Instead the matrix

has to be numerically solved for each frequency

used in the filter design procedure.

5 FILTER DESIGN EXAMPLES

In the following we will discuss the design of a

two-periodic time-varying FIR filter, the

compensation of a two-channel time-interleaved

DAC, and the design of a Farrow filter.

5.1 FIR filter design

For the design example we have assumed atwo-periodic ZOH signal with time offsets of r

0= 0

and r 1= 0.0391. To approximate the ideal

frequency responses given by Eq. (38) we use

causal finite-impulse response (FIR) filters with

transfer functions

Gn

ae j ( ) = gn

a[l ]e

j l

l =0

L

(40)

and approximate the ideal frequency responses in

the minimax (Chebychev) sense, i.e.,

minGne j ( )Gn

ae j ( )

D (41)

where D

is the definition domain of and

frequencies above D

belong to the “don't care”

band. We have designed filters of order L =17 ,

with = 8.5, and D= (0.7 ,K,0.7 ) by

solving the approximation problem with the Matlab

optimization toolbox CVX [15].




0 0.1 0.2 0.3 0.4 0.5−140

−120

−100

−80

−60

−40

−20

0

E 0

( e

j ω ) ( d B )


Figure 6: Deviation from the ideal overall

frequency response.

0 0.1 0.2 0.3 0.4 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

E 1

( e j ω ) ( d B )


Figure 7: Attenuation of the spurious images.

For the specified problem we have obtained a

maximum approximation error of 61.1 dB in the

filter design of G0

a(e

j

) and 60.4 dB for G1

a(e

j

).

In Fig. 6 we see the impact of the approximation

error on

E 0e j ( ) =

(

F 0e j ( ) e j (42)

i.e., the difference between the average frequency

response and an ideal delay, and in Fig. 7 on the

residual images

E 1e j ( ) =

(

F 1e j ( ) . (43)

Within the definition band D

, the approximation

error of an ideal delay is less than -98 dB and the

spectral images are attenuated by at least -60 dB.

To further test the filter design, a discrete-time

multitone signal with frequencies [0.0581, 0.1162,

0.1743, 0.2324, 0.2905, 0.3486] 1/T has been used

as an input to the nonuniform ZOH. Without any

compensation we have obtained the energy density

spectrum shown in Fig. 8. We see the sin( x ) / x

0 0.5 1 1.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0


B c )


Figure 8: Output of a nonuniform ZOH with

period M=2.

0 0.5 1 1.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0


B c )


Figure 9: Precompensated output of anonuniform ZOH with period M=2.

output spectrum and the additional image spectra

due to the nonuniform ZOH signal. The largest

unwanted spur in the fundamental band, i.e., the

band between 0 and 0.5, is about -29 dBc. The

energy density spectrum with compensation is

shown in Fig. 9. Within the fundamental band the

unwanted spectral images are considerably reduced.

The largest spur is about -61 dBc, which is an

improvement of 32 dB compared to the

uncompensated case. A higher attenuation of the

images is achievable by increasing the filter order.When comparing the out-of-band energy of

Fig. 8 and Fig. 9 we recognize that the out-of-band

energy in Fig. 9 is slightly increased. This minor

amplification of the out-of-band energy, however,

should not change the design requirements of the

analog reconstruction filter significantly.

5.2 Time-interleaved DAC

We have implemented a two-channel time-

interleaved digital-to-analog converter that

produces two-periodic nonuniform ZOH signals [8].

Therefore, we have used a XILINX Virtex-4 FPGA

evaluation platform [16] to generate the digitalsignal and two high-speed DACs from Analog




Devices (AD9734) [17] to convert the digital

signals into the analog domain.

An 800 MHz differential clock generator has

generated the 180 degrees phase-shifted clock

signals for the two DACs, where the differential

outputs have been used as single ended clock

signals. Thus, the overall sampling rate of the two-channel time-interleaved DAC is 1600 MS/s. The

two analog output signals have been combined with

a power combiner and measured with an Agilent

54855A 6 GHz scope with a sampling frequency of

20 GS/s.

For the presented measurements we have

generated a single sinusoid with a frequency of

300 MHz and have identified the time offsets as

r 0= 0 and r

1= 0.12 . Along with the identified time

offsets a filter, which compensates the digital signal

as depicted in Fig. 4, has been designed according

to Eq. (41). The digitally compensated and

converted signal has been measured with the high-speed sampling scope. The results of the

measurements are shown in Fig. 10. The mirror

images at 500 MHz have been reduced by about

20 dB. Due to uncertainties in the time offset

estimation, a better reduction has not been possible.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.42.4−

80

−60

−40

−20

0

Frequency (GHz) E n e

r g y D e n s i t y S p e c t r u m ( d

B c )

uncompensated

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−80

−60

−40

−20

0


B c )

Frequency (GHz)

compensated

Figure 10: Measurements for the

uncompensated and the precompensated

sinusoid.

5.3 Farrow filter designAs shown in the last section, we can design a

time-varying filter consisting of two impulse

responses to compensate two-periodic nonuniform

ZOH signals with given deviations. In some

applications it is necessary to redesign the

compensation filter from time to time, since the

time offsets r n

are subject to temperature changes,

aging effects, and voltage supply variations.

Therefore, the previously introduced filter design

process might become too complex for certain

applications. For the two-periodic case, the solution

to this problem is the adaptation of a Farrow

filter [18], which reduces the redesign complexity

significantly.

By defining r n+1 r

n= (1)

n

and r n+1+ r

n= ,

we can rearrange Eq. (20) as

H n (e

j ) = A

n (e j

, )e j 1+ ( )

1

2 (44)

with

An (e j

, ) =

sin 1+ 1( )n ( )

1

2

1

2

. (45)

With these definitions, the frequency responses

An(e

j , ) depend on one additional spectral

parameter as required for Farrow-based filter

designs. Furthermore, all frequency responses

H n (e

j

) share a common factor exp(

j 1+

( ) / 2) .Hence, we can conclude from Eq. (18) and Eq. (20)

that the factor exp( j 1+ ( ) / 2) shifts the entire

input signal by (1+ ) / 2 in time, but does not

produce spurious images. Since we are only

interested in compensating the spurious images, we

can modify our perfect reconstruction condition

given by Eq. (34) for the two-periodic case as

(

F l e j l

2

M

= e

j + 1+ ( )1

2

, for l = 0

0, for l =1

(46)

By using Eq. (44) and Eq. (46) and following the

derivations of the last section we obtain the

compensation filters as

G0

e j , ( )

G1

e j , ( )

=

A1

e j + ( )

, ( )P e j ( )

A0

e j + ( )

, ( )P e

j ( )

e j . (47)

The ideal time-varying compensation filter depends

on the frequency and on an additional spectral parameter . Hence, these filters can be designed

and implemented as Farrow f ilters.

A Farrow filter has an impulse response of

gn

a[ l , ]= cn , p[ l ]

p

p=0

P1

(48)

with

Gn

ae j

, ( ) = gn

al , [ ]

l =0

L

e j l . (49)




In general, we have to design two sets of

Farrow coefficients c0, p[ l ] and c

1, p[ l ] for our two-

periodic compensation filter. However, we can

further simplify the filter design by exploiting the

even symmetry regarding , i.e.,

A0 e j , ( ) = A1 e j , ( ) (50)

which can be verified with Eq. (45). In consequence,

we can show that

P e j

, ( ) = P e j

, ( ) (51)

and can conclude that the transfer functions of the

two-periodic time-varying filter are even functions

regarding the spectral parameter

G0 e

j

,

( )=

G1 e

j

,

( ) . (52)

This property can be exploited for the filter design

by choosing a definition domain for D

that is

symmetric around zero, i.e., D= ( max

,K, max).

For such a choice, the filter design will lead to two

transfer functions related by

G0

ae j

, ( ) =G1

ae j

, ( ) (53)

and can be described in general as

Gn

a e j , ( ) =G0

a e j ,1( )n

( ) . (54)

Accordingly, we only have to design one set of

filter coefficients c0, p[l ] and can efficiently

implement the compensation filter as shown in

Fig. 11.

s[n]

c0,1[l]c0,2[l] c0,0[l]

(−1)nλ(−1)nλ

Gan(e

jω)x[n]

Figure 11: Proposed modified Farrow filter for

P=3.

Furthermore, G0 (e j

, ) consists of a real zero-

phase response A0 (e

j + ( ), ) /P(e

j ) and a linear-

phase response exp( j ) , which can be chosen

arbitrarily. By choosing a delay such that the

compensation filter will exhibit symmetrical or

anti-symmetrical impulse responses, the number of

coefficient multipliers can be basically halved [11,

Chap. 6].

For the design example we have chosen an FIR

Farrow filter of order P = 3 with subfilters of order

L=

17 and a delay of =

8.5, i.e., subfilters with9 multipliers if we exploit the linear-phase

property, with D= (0.7 ,K,0.7 ). The range of

possible time offsets has been D= (0.05,K,0.05) .

The approximation problem was solved in the

Chebyshev sense ( L

norm), i.e.,

minG0e j , ( )G0

a e j , ( )

D , D (55)

where D

is the definition domain of the spectral

parameter , by again using Matlab and the

optimization toolbox CVX. Design procedures can

be found for example in [19].

0 0.1 0.2 0.3 0.4 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

E 0

( e j ω ) ( d B )


λ=±0.05

λ=±0.0391

λ=±0.0278

λ=±0.0167

λ=±0.0056

Figure 12: Deviation from the ideal overall

frequency response for different .

0 0.1 0.2 0.3 0.4 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

E 1

( e j ω ) ( d B )


λ=±0.05

λ=±0.0391

λ=±0.0278

λ=±0.0167

λ=±0.0056

Figure 13: Attenuation of the spurious images

for different .

For the specified problem we have obtained a

maximum approximation error of 55.3 dB in thefilter design of G

0

a(e

j , ) . In Fig. 12 we see the




impact of the approximation error due to the filter

design on realizing an ideal delay, i.e.,

E 0e j ( ) =

(

F 0e j ( )e

j + 1+ ( )1

2

(56)

and in Fig. 13 on the residual images as in Eq. (43)

for various values of . Within the definition band,

the approximation error of an ideal delay is less

than -70 dB and the spectral images are attenuated

by at least -63 dB. These approximation errors are

comparable to the values for the fixed FIR filter;

however, for the given specification, the flexible

Farrow-based solution needs at least 3 times more

multipliers in its implementation. Hence, we can

avoid the online filter design and gain flexibility,

but have to pay for it with higher implementation

costs.

Finally, we have simulated two-periodic ZOHsignals with time offsets r

0= 0 and r

1= 0.0391

driven by a digitally generated multi-tone signal as

in Sec. 5.1. The energy density spectra of the

uncompensated and the compensated output of the

ZOH are shown in Fig. 8 and Fig. 14, respectively.

In the compensated spectrum, the spurs are reduced

to -63 dBc.

Figure 14: Output of a two-periodic ZOH

precompensated by a Farrow filter.

6 CONCLUDING REMARKS

We have introduced the design of a discrete-time

time-varying filter to compensate the spurious in-

band image spectra caused by M-periodic

nonuniform ZOH signals. A general filter design

procedure is derived and the filter design for the

two-channel case is demonstrated by two examples.

Furthermore, for the two-channel case, we have

extended the design procedure to Farrow filters,

which allows for compensating a wide range of

possible time offsets by the simple adjustment of

some multipliers.

ACKNOWLEDGEMENT

Christian Vogel was supported by the Austrian

Science Fund FWF's Erwin Schrödinger Fellowship

J2709-N20.

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